Complete the proof.
Given: ∠DCE≅∠CDE≅∠DEC; ∠CDB≅∠CBD
Prove: BC≅DE
a. DE≅EC≅CD because ___?
b. BC≅DC because ___?
c. ∴ BC≅DE because ___?
To complete the proof, we need to provide reasons for each step. Here's the proof:
a. DE≅EC≅CD because the given statement states that ∠DCE≅∠CDE≅∠DEC. Since angle ≅ side ≅ side, it implies that the corresponding segments are congruent.
b. BC≅DC because the given statement states that ∠CDB≅∠CBD. Again, using the same reasoning as before, ≅ angles imply ≅ sides.
c. ∴ BC≅DE because BC and DC are both congruent to the same segment CD based on step b, and CD is congruent to DE based on step a. Therefore, BC and DE are congruent to the same segment, making them congruent to each other.
To complete the proof, we will use the given information to establish the equality of segments and angles step by step.
a. DE≅EC≅CD because of the angle bisector theorem.
Explanation: From the given information, we know that ∠DCE≅∠CDE≅∠DEC. These angles are all congruent, meaning they have equal measures. By the angle bisector theorem, when the three angles are congruent, the segments opposite those angles are also congruent. Therefore, we can conclude that DE≅EC≅CD.
b. BC≅DC because of the angle bisector theorem.
Explanation: From the given information, we also know that ∠CDB≅∠CBD. These angles are congruent, meaning they have equal measures. Applying the angle bisector theorem once again, we can conclude that the segments opposite those angles are also congruent. Hence, BC≅DC.
c. ∴ BC≅DE because of the transitive property of equality.
Explanation: Using the transitive property of equality, we can combine the previous information. We have BC≅DC (from step b) and DE≅EC≅CD (from step a). Since DC is equal to both DE and CD, we can use the transitive property to state that BC is also equal to DE. Therefore, we can conclude that BC≅DE.
By establishing the equality of BC and DE using the given information and the angle bisector theorem, we have completed the proof.